Remark 3.4 For any \(i\in I\) , Lusztig (Section 1.2.13 in [1]}) introduced a derivation \({r_i}:\mathbf {U}^-\rightarrow \mathbf {U}^-\)  such that
\({r^m_i}(xy)=\sum _{t=0}^{m}v^{(ti,\nu -(m-t)i)+t(m-t)}\frac{[m]_{v}!}{[t]_{v}![m-t]_{v}!}{r_i^t}(x){r_i^{m-t}}(y)\) 
